On nonlocal fractional boundary value problems

This paper studies a new class of nonlocal boundary value problems of nonlinear differential equations of fractional order. We extend the idea of a three-point nonlocal boundary condition (x(1) = αx(η), α e R, 0 < η < 1) to a nonlocal strip condition of the form: x(1) = η ∫vτ x(s)ds, 0 < v < τ < 1. In fact, this strip condition corresponds to a continuous distribution of the values of the unknown function on an arbitrary finite segment of the interval. In the limit v → 0, τ → 1, this strip condition takes the form of a typical integral boundary condition. Some new existence and uniqueness results are obtained for this class of nonlocal problems by using standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also discussed.